Firm Behavior under Profit Maximization Monopoly Bertrand Price Competition
Monopoly A monopoly solves Max p(q)q-c(q) –q is quantity. –c(q) is cost of producing quantity q. –p(q) is price (price depends upon output). FOC yields p(q)+p(q)q=c(q). This is also Marginal Revenue=Marginal Cost.
Example (from Experiment) We had quantity q=15-p. While we were choosing prices. This is equivalent (in the monopoly case) to choosing quantity. r(q)= q*p(q) where p(q)=15-q. Marginal revenue was 15-2q. We had constant marginal cost of 3. Thus, c(q)=3*q. Profit=q*(15-q)-3*q What is the choice of q? What does this imply about p?
Bertrand (1883) price competition. Both firms choose prices simultaneously and have constant marginal cost c. Firm one chooses p1. Firm two chooses p2. Consumers buy from the lowest price firm. (If p1=p2, each firm gets half the consumers.) An equilibrium is a choice of prices p1 and p2 such that –firm 1 wouldnt want to change his price given p2. –firm 2 wouldnt want to change her price given p1.
Bertrand Equilibrium Take firm 1s decision if p2 is strictly bigger than c: –If he sets p1>p2, then he earns 0. –If he sets p1=p2, then he earns 1/2*D(p2)*(p2-c). –If he sets p1 such that c<p1<p2 he earns D(p1)*(p1-c). For a large enough p1 that is still less than p2, we have: –D(p1)*(p1-c)>1/2*D(p2)*(p2-c). Each has incentive to slightly undercut the other. Equilibrium is that both firms charge p1=p2=c. Not so famous Kaplan & Wettstein (2000) paper shows that there may be other equilibria with positive profits if there arent restrictions on D(p).
Bertrand Game Firm B Firm A £9 18 35.75 0 18 0 17.88 £8.50 £9 £8.50 Marginal cost= £3, Demand is 15-p. The Bertrand competition can be written as a game. For any price> £3, there is this incentive to undercut. Similar to the prisoners dilemma. 35.75 17.88
Sample result: Bertrand Game Two Firms Fixed Partners Two Firms Random Partners Five Firms Random Partners
Cooperation in Bertrand Comp. A Case: The New York Post v. the New York Daily News January 1994 40¢ 40¢ February 1994 50¢ 40¢ March 1994 25¢ (in Staten Island) 40¢ July 1994 50¢ 50¢
What happened? Until Feb 1994 both papers were sold at 40¢. Then the Post raised its price to 50¢ but the News held to 40¢ (since it was used to being the first mover). So in March the Post dropped its Staten Island price to 25¢ but kept its price elsewhere at 50¢, until News raised its price to 50¢ in July, having lost market share in Staten Island to the Post. No longer leader. So both were now priced at 50¢ everywhere in NYC.
Collusion If firms get together to set prices or limit quantities, what would they choose? As in your experiment. D(p)=15-p and c(q)=3q. Price Max p (p-3)*(15-p) What is the choice of p? This is the monopoly price and quantity! Max q1,q2 (15-q1-q2)*(q1+q2)-3(q1+q2).
Graph of total profit: (15-price)(price-3) Profit Price Maximum is price=9 With profit 36.
Collusion by Repeated Interaction Let us say that firms have a discount factor of B. If each make 18 each period. How much is the present value? The one period undercutting gains is close to 18. The other firm can punish under-cutters by causing zero profit from then on. A firm will not cheat only if the punishment is worse than the gains. For what values of B will the firm not cheat? 18B/(1-B)>=18 (or B>=1/2).
Anti-competitive practices. In the 80s, Crazy Eddie said that he will beat any price since he is insane. Today, many companies have price-beating and price-matching policies. A price-matching policy is simply if you (a customer) can find a price lower than ours, we will match it. A price-beating policy is that we will beat any price that you can find. (It is NOT explicitly setting a price lower or equal to your competitors.)
Price Matching/Price Beating They seem very much in favor of competition: consumers are able to get the lower price. In fact, they are not. By having such a policy a stores avoid loosing customers and thus are able to charge a high initial price (yet another paper by this Kaplan guy).
Price-matching Marginal cost is 3 and demand is 15-p. There are two firms A and B. Customers buy from the lowest price firm. Assume if both firms charge the same price customers go to the closest firm. What are profits if both charge 9? Without price matching policies, what happens if firm A charges a price of 8? Now if B has a price matching policy, then what will Bs net price be to customers? B has a price-matching policy. If B charges a price of 9, what is firm As best choice of a price. If both firms have price-matching policies and price of 9, does either have an incentive to undercut the other?
Price-Matching Policy Game Firm B Firm A £9 18 17.88 18 17.88 £8.50 £9 £8.50 Marginal cost= £3, Demand is 15-p. If both firms have price-matching policies, they split the demand at the lower price. The monopoly price is now an equilibrium! 17.88
Rule of thumb prices Many shops use a rule of thumb to determine prices. Clothing stores may set price double their costs. Restaurants set menu prices roughly 4 times costs. Can this ever be optimal? q=Ap є (p=(1/A) 1/є q 1/є ) where -1> є Notice in this case that p(q)+p(q)q=((1+є)/ є)p(q). If marginal cost is constant, then p= є/(1+є)mc for any mc. There is a constant mark-up percentage! Notice that (dq/q)/(dp/p)= є. What does є represent?
Homework El Al and British Air are competing for passengers on the Tel Aviv- Heathrow route. Assume marginal cost is 4 and demand is Q = 18 P. –If they choose prices simultaneously, what will be the Bertrand equilibrium? –If they can collude together and fix prices, what would they charge. –In practice with such competition under what conditions would you expect collusion to be strong and under what conditions would you expect it to be weak. –Under what conditions should the introduction of BMI (another airline) affect prices? –If the game is infinitely repeated, under what discount factor B would full collusion be obtainable according to standard game theory.